2%3Dx%2B2%2Fx%2B4.jpeg "(x+1/x+2)2=x+2/x+4")
Solving the Equation: (x+1/x+2)2 = x+2/x+4
This equation presents a unique challenge due to the presence of fractions and squares. Let's break down the steps to find the solution:
1. Simplify the Equation
Begin by simplifying the left side of the equation. Expand the square:
(x+1/x+2)² = (x+1/x+2) * (x+1/x+2)
Using the FOIL method (First, Outer, Inner, Last):
= x² + (2x/x+2) + (x/x+2) + 1/(x+2)²
Combining like terms:
= x² + (3x/x+2) + 1/(x+2)²
Now the equation becomes:
x² + (3x/x+2) + 1/(x+2)² = x+2/x+4
2. Eliminate Fractions
To get rid of the fractions, multiply both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM is (x+2)²(x+4):
(x+2)²(x+4) * [x² + (3x/x+2) + 1/(x+2)²] = (x+2)²(x+4) * (x+2/x+4)
This simplifies to:
(x+2)²(x+4)x² + 3x(x+2)(x+4) + (x+4) = (x+2)³
3. Expand and Rearrange
Expand the terms on both sides of the equation and move all terms to one side:
x⁵ + 8x⁴ + 20x³ + 16x² - 8x - 8 = 0
4. Solve for x
The resulting equation is a quintic equation, which generally does not have a simple analytical solution. Therefore, we would need to resort to numerical methods to find the approximate values of x. These methods include:
- Graphical Solution: Plotting the function f(x) = x⁵ + 8x⁴ + 20x³ + 16x² - 8x - 8 and identifying the x-intercepts.
- Numerical Methods: Using iterative algorithms like Newton-Raphson method to approximate the roots.
Important Note: It is crucial to check the validity of the solutions obtained from these methods by substituting them back into the original equation to ensure they satisfy the equation.
Conclusion
While a closed-form solution to the equation (x+1/x+2)² = x+2/x+4 is not readily available, we can use numerical methods to approximate the solutions. Remember to check the validity of the solutions obtained.